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A nodally bound-preserving composite discontinuous Galerkin method on polytopic meshes

Abdolreza Amiri, Gabriel R. Barrenechea, Emmanuil H. Georgoulis, Tristan Pryer

TL;DR

This work addresses the challenge of enforcing physical bounds for elliptic problems on flexible polytopic meshes by introducing a nodally bound-preserving composite DG method. The approach augments a baseline interior-penalty DG discretisation with a simplicial submesh and a nonlinear bound-preserving projection $\mathcal{E}^+$, forming a semilinear form $a_h(u_H;v_H)=a_{DG}(\mathcal{E}^+(u_H),v_H)+s(\mathcal{E}^-(u_H),v_H)$ that preserves the discrete bounds at chosen nodes while maintaining the original DG accuracy. A stabilising kernel $s$ ensures well-posedness via monotonicity, and a novel nodally bound-preserving interpolant yields a priori error bounds; the penalty parameter remains governed by the polytopic mesh rather than the submesh. Theoretical results include existence/uniqueness and optimal convergence, complemented by extensive numerical experiments showing robustness to boundary/interior layers and excellent agreement with standard DG where bounds are satisfied. Overall, the method enables accurate, bound-respecting simulations on general polygonal/polyhedral meshes with arbitrary-order accuracy and without increasing global degrees of freedom.

Abstract

We introduce a nodally bound-preserving Galerkin method for second-order elliptic problems on general polygonal/polyhedral, henceforth collectively termed as \emph{polytopic}, meshes. Starting from an interior penalty discontinuous Galerkin (DG) formulation posed on a polytopic mesh, the method enforces preservation of \emph{a priori} prescribed upper and lower bounds for the numerical solution at an arbitrary number of user-defined points \emph{within} each polytopic element. This is achieved by employing a simplicial submesh and enforcing bound preservation at the submesh nodes via a nonlinear iteration. By construction, the submeshing procedure preserves the order of accuracy of the DG method, \emph{without} introducing any additional global numerical degrees of freedom compared to the baseline DG method, thereby, falling into the category of composite finite element approaches. A salient feature of the proposed method is that it automatically reverts to the standard DG method on polytopic meshes when no prescribed bound violation occurs. In particular, the choice of the discontinuity-penalisation parameter is independent of the submesh granularity. The resulting composite method combines the geometric flexibility of polytopic meshes with the accuracy and stability of discontinuous Galerkin discretisations, while rigorously guaranteeing bound preservation. The existence and uniqueness of the numerical solution is proven. A priori error bounds, assuming sufficient regularity of the exact solution are shown, employing a non-standard construction of discrete nodally bound-preserving interpolant. Numerical experiments confirm optimal convergence for smooth problems and demonstrate robustness in the presence of sharp gradients, such as boundary and interior layers.

A nodally bound-preserving composite discontinuous Galerkin method on polytopic meshes

TL;DR

This work addresses the challenge of enforcing physical bounds for elliptic problems on flexible polytopic meshes by introducing a nodally bound-preserving composite DG method. The approach augments a baseline interior-penalty DG discretisation with a simplicial submesh and a nonlinear bound-preserving projection , forming a semilinear form that preserves the discrete bounds at chosen nodes while maintaining the original DG accuracy. A stabilising kernel ensures well-posedness via monotonicity, and a novel nodally bound-preserving interpolant yields a priori error bounds; the penalty parameter remains governed by the polytopic mesh rather than the submesh. Theoretical results include existence/uniqueness and optimal convergence, complemented by extensive numerical experiments showing robustness to boundary/interior layers and excellent agreement with standard DG where bounds are satisfied. Overall, the method enables accurate, bound-respecting simulations on general polygonal/polyhedral meshes with arbitrary-order accuracy and without increasing global degrees of freedom.

Abstract

We introduce a nodally bound-preserving Galerkin method for second-order elliptic problems on general polygonal/polyhedral, henceforth collectively termed as \emph{polytopic}, meshes. Starting from an interior penalty discontinuous Galerkin (DG) formulation posed on a polytopic mesh, the method enforces preservation of \emph{a priori} prescribed upper and lower bounds for the numerical solution at an arbitrary number of user-defined points \emph{within} each polytopic element. This is achieved by employing a simplicial submesh and enforcing bound preservation at the submesh nodes via a nonlinear iteration. By construction, the submeshing procedure preserves the order of accuracy of the DG method, \emph{without} introducing any additional global numerical degrees of freedom compared to the baseline DG method, thereby, falling into the category of composite finite element approaches. A salient feature of the proposed method is that it automatically reverts to the standard DG method on polytopic meshes when no prescribed bound violation occurs. In particular, the choice of the discontinuity-penalisation parameter is independent of the submesh granularity. The resulting composite method combines the geometric flexibility of polytopic meshes with the accuracy and stability of discontinuous Galerkin discretisations, while rigorously guaranteeing bound preservation. The existence and uniqueness of the numerical solution is proven. A priori error bounds, assuming sufficient regularity of the exact solution are shown, employing a non-standard construction of discrete nodally bound-preserving interpolant. Numerical experiments confirm optimal convergence for smooth problems and demonstrate robustness in the presence of sharp gradients, such as boundary and interior layers.

Paper Structure

This paper contains 12 sections, 8 theorems, 132 equations, 7 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Model problem and its discretisation by the discontinuous Galerkin method
  3. Model problem
  4. Admissible partitions and finite element spaces
  5. Interior penalty discontinuous Galerkin method
  6. A nodally bound-preserving composite discontinuous Galerkin method
  7. Well-posedness
  8. Bound-preserving best approximation
  9. A priori error analysis
  10. Implementation and matrix structure
  11. Numerical experiments
  12. Conclusion

Key Result

Lemma 2.4

Let $F\subset \partial K_+\cap\partial K_-$ be a generic face shared by two elements $K_+,K_-\in{\mathcal{P}}$. When $F\subset\partial\Omega$, we set $K_-=\emptyset$. Defining for every face $F\subset\Gamma_{{\mathcal{P}}}$, we have

Figures (7)

  • Figure 1: Representative polytopic mesh and two associated triangular submeshes at different quadrature levels.
  • Figure 2: Structure of the transformation matrices $\mathbf{O}$ and $\mathbf{Q}$ on Mesh \ref{['M3']} with $\mathbb{P}_{1}$ elements.
  • Figure 3: Structure of the stiffness matrices $\mathbf{A}_{\rm DG}$, $\mathbf{A}_{\#}$ and $\mathbf{A}_{{\mathcal{P}}}$ on Mesh \ref{['M3']} with $\mathbb{P}_{1}$ elements.
  • Figure 4: Discrete solution $\mathcal{E}^{+}(u_{H})$ for Example \ref{['Example1']} with $c=8$ on a mesh with $\#{\mathcal{P}}=512$ using $\mathbb{P}_{1}$ elements.
  • Figure 5: Approximate solutions $\mathcal{E}^{+}(u_{H})$ for Example \ref{['Example22']} using $\mathbb{P}_{2}$ on the finest mesh ($\#{\mathcal{P}}=1024$).
  • ...and 2 more figures

Theorems & Definitions (22)

  • Lemma 2.4: Coercivity of $a_{DG}(\cdot,\cdot)$
  • proof
  • Remark 3.1: Properties of the recovery operator
  • Remark 3.2: Properties of the bound-preserving DG method
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • Lemma 3.5
  • proof
  • Lemma 3.6
  • ...and 12 more